Chapter 9 Differential Geometry 9.1. Theory of Curves 9.1.1. Plane Curves 9.1.1-1. Regular points of plane curve. AplanecurveΓ in a Cartesian coordinate system can be defined by equations in the following form: Explicitly, y = f (x). (9.1.1.1) Implicitly, F (x, y)=0. Parametrically, x = x(t), y = y(t). (9.1.1.2) In vector form, r = r(t), where r(t)=x(t)i + y(t)j . In a polar coordinate system, the curve is usually given by the equation r = r(ϕ), where the relationship between Cartesian and polar coordinates is given by formulas x = r cos ϕ and y = r sin ϕ. Remark. The explicit equation (9.1.1.1) can be obtained from the parametric equations (9.1.1.2) if the abscissa is taken for the parameter: x = t, y = f(t). A point M(x(t), y(t)) is said to be regular if the functions x(t)andy(t) have continuous first derivatives not simultaneously equal to zero in a sufficiently small neighborhood of this point. For implicitly defined functions, a point M(x, y)issaidtoberegular if grad F = ∇F ≠ 0 at this point. If acurve is given parametrically (9.1.1.2), then thepositive sense is defined on this curve, i.e., the direction in which the point M(x(t), y(t)) of the curve moves as the parameter t increases. If the curve is given explicitly by (9.1.1.1), then the positive sense corresponds to the direction in which the abscissa increases (i.e., moves from left to right). In a polar coordinate system, the positive sense corresponds to the direction in which the angle ϕ increases (i.e., the positive sense is counterclockwise). If s is the curve length from some constant point M 0 to M, then the infinitesimal length increment of the arc M 0 M is approximately expressed by the formula for the arc length 367 368 DIFFERENTIAL GEOMETRY differential ds; i.e., the following formulas hold: Δs ≈ ds = 1 +(y x ) 2 dx, if the curve is given explicitly, Δs ≈ ds = (x t ) 2 +(y t ) 2 dt, if the curve is given parametrically, Δs ≈ ds = r 2 +(r ϕ ) 2 dϕ, for a curve in the polar coordinate system. Example 1. The arc length differential of the curve y =cosx has the form ds = √ 1 +sinxdx. Example 2. For the semicubical parabola x = t 2 , y = t 3 , the arc length differential is equal to ds = t √ 4 + 9t 2 dt. Example 3. For the hyperbolic spiral r = a/ϕ for r > 0, the arc length differential is equal to ds = a 1 + ϕ 2 /ϕ 2 dϕ. 9.1.1-2. Tangent and normal. The tangent to a curve Γ at a regular point M 0 is defined to be the straight line that is the limit position of the secant M 0 M 1 as the point M 1 approaches the point M 0 ;thenormal is defined to be the straight line passing through M 1 and perpendicular to the tangent (Fig. 9.1). normal tangent M M 1 0 Figure 9.1. Tangent and normal. At each regular point M(x 0 , y 0 )=M(x(t 0 ), y(t 0 )), the curve Γ has a unique tangent given by one of the equations (depending on how the curve is defined) y – y 0 = y x (x – x 0 ), if the curve is given explicitly, F x (x – x 0 )+F y (y – y 0 )=0, if the curve is given implicitly, y – y 0 y t = x – x 0 x t , if the curve is given parametrically, r = r 0 + λr t , if the curve is given in vector form, where r 0 is the position vector of the point M 0 , λ is an arbitrary parameter, and all derivatives are evaluated at x = x 0 , y = y 0 ,andt = t 0 . The slope of the tangent is determined by the angle α between the positive direction of the OX-axis and the positive direction of the tangent (Fig. 9.2a). The slope of the tangent (and the angle α) is determined by the formulas tan α = y x =– F x F y = y t x t . 9.1. THEORY OF CURVES 369 M ()a ()b M O α θ X Y Figure 9.2. Slope of the tangent. If a curve is given in the polar coordinate system, then the slope of the tangent is determined by the angle θ between the direction of the position vector r = OM and the positive direction of the tangent (Fig. 9.2b). The angle θ is determined by the formula tan θ = r ϕ r . The normal at each regular point M (x 0 , y 0 )=M(x(t 0 ), y(t 0 )) is given, depending on the method for defining the curve, by the equations y – y 0 =– x – x 0 y x , if the curve is given explicitly, x – x 0 F x = y – y 0 F y , if the curve is given implicitly, (y – y 0 )y t +(x – x 0 )x t = 0, if the curve is given parametrically, where all the derivatives are evaluated at x = x 0 , y = y 0 ,andt = t 0 . The positive sense of the tangent coincides with the positive sense of the curve at the point of tangency; and the positive sense of the normal can in some way be made consistent with the positive sense of the tangent; for example, it can be obtained from the positive sense of the tangent by counterclockwise rotation around M by an angle of 90 ◦ (Fig. 9.3). The point M divides the tangent and the normal into positive and negative half-lines. M Figure 9.3. Positive sense of tangent. Example 4. Let us find the equations of the tangent and the normal to the parametrically given semicubical parabola x = t 2 , y = t 3 at the point M 0 (1, 1), t = 1. The equation of the tangent, y – t 3 3t 2 = x – t 2 2t or y = 3 2 tx – 1 2 t 3 , at the point M 0 (1, 1)is y = 3 2 x – 1 2 . 370 DIFFERENTIAL GEOMETRY The equation of the normal, 2t(x – t 2 )+3t 2 (y – t 3 )=0 or 2x + 3ty = t 2 (2 + 3t 2 ), at the point M 0 (1, 1) is (Fig. 9.4a) 2x + 3y = 5. M yx= 31 22 X 1 1 1 2 2 3 3 O 234 Y 2+3=5xy 0 M O 22 2 2 4 4 X Y (a)(b) xy+=8 yx= 0 Figure 9.4. Tangents and normals to the semicubical parabola (a) and to the circle (b). Example 5. Let us find the equation of the tangent and the normal to the circle x 2 + y 2 = 8 at the point M 0 (2, 2). We write the equation of the circle as F (x,y)=0: x 2 + y 2 – 8 = 0, i.e., F (x,y)=x 2 + y 2 – 8. Obviously, we obtain F x = 2x, F y = 2y. The equation of the tangent is 2x 0 (x – x 0 )+2y 0 (y – y 0 )=0, or, taking into account the original equation of the circle, xx 0 + yy 0 = 8. At the point M 0 (2, 2), we have x + y = 4. The equation of the normal is x – x 0 2x 0 = y – y 0 2y 0 , or y = y 0 x 0 x. At the point M 0 (2, 2)(Fig.9.4b), we have y = x. 9.1. THEORY OF CURVES 371 M O 1 1 X π Y 0 yx= yx= yx=cos π π 2 2 Figure 9.5. The tangent and the normal to the curve y =cosx. Example 6. Let us find the equations of the tangent and the normal to the curve y =cosx at the point M 0 (π/2, 0). The equation of the tangent is y –cosx 0 =–sinx 0 (x – x 0 )ory =cosx 0 –sinx 0 (x – x 0 ). At the point M 0 (π/2, 0), we have y = π 2 – x. The equation of the normal is y –cosx 0 =– x – x 0 –sinx 0 or y =cosx 0 + x – x 0 sin x 0 . At the point M 0 (π/2, 0) (Fig. 9.5), we have y = x – π 2 . 9.1.1-3. Singular points. A point is said to be singular if it is not regular. Implicit equations of the form F (x, y)=0 are used as a rule to find singular points of a curve and analyze their character. At any singular point M 0 (x 0 , y 0 ), both partial derivatives of the function F (x, y) are zero: F x (x 0 , y 0 )=0 and F y (x 0 , y 0 )=0. If both first partial derivatives are zero at M 0 and simultaneously at least one of the second derivatives F xx , F xy ,andF yy is nonzero, then M 0 is called a double point. This is the most widely known case of singular points. If both first partial derivatives and simultaneously all second partial derivatives are zero at M 0 but not all third partial derivatives are zero at M 0 , then the point M 0 is said to be triple. In general, if all partial derivatives of F(x, y)upto order n – 1 inclusive are zero at M 0 but at least one of the nth derivatives is nonzero at M 0 , then the point M 0 is called an n-fold singular point.Atann-fold singular point M 0 ,the curve has n tangents, some of which may coincide or be imaginary. For example, for a double singular point M 0 , the slopes λ = y x of the two tangents at this point are the roots of the quadratic equation F yy (x 0 , y 0 )λ 2 + 2F xy (x 0 , y 0 )λ + F xx (x 0 , y 0 )=0.(9.1.1.3) The roots of equation (9.1.1.3) depend on the sign of the expression Δ = F xx F xy F yx F yy = F xx F yy – F 2 xy , where the second derivatives are evaluated at the point M 0 (x 0 , y 0 ). If Δ > 0, then the roots of the quadratic equation (9.1.1.3) are complex conjugate. In this case, a sufficiently small neighborhood of M 0 (x 0 , y 0 ) does not contain any other points of the curve except for M 0 itself. Such a point is called an isolated point. 372 DIFFERENTIAL GEOMETRY Example 7. The curve y 2 + 4x 2 – x 4 = 0 has the isolated point (0, 0)(Fig.9.6a). O X 1 12 5 5 2 Y ()a ()b O X Y Figure 9.6. Examples of the isolated point (a) and the node (b). If Δ < 0, then the quadratic equation (9.1.1.3) has two distinct real roots. In this case, there are two branches of the curve passing through the point M 0 (x 0 , y 0 ); these branches have distinct tangents whose directions are just determined by equation (9.1.1.3). Such a point is called a node (a point of self-intersection). Example 8. The point (0, 0)ofthecurve y 2 – x 2 = 0 is the node (0, 0) (Fig. 9.6b). If Δ = 0, then the roots of the quadratic equation (9.1.1.3) coincide. In this case, the singular point of the curve is either isolated or characterized by the fact that all branches approaching the singular point M 0 have a common tangent at this point: 1. Cusps of the first kind are points approached by two branches of the curve that have a common tangent at this point and lie on the same side of the common normal and on opposite sides of the common tangent. 2. Cusps of the second kind are points approached by two branches of the curve that have a common tangent at this point and lie on the same side of the common normal and on the same side of the common tangent. 3. Points of osculation are points at which the curve is tangential to itself. Example 9. For the curve (cissoid of Diocles) (2a – x)y 2 – x 3 = 0, shown in Fig. 9.7a, the origin is a cusp of the first kind, which is clear from the explicit equation y = x 3 2a – x of the curve. For x < 0,noy satisfy the equation, and for x > 0 the values y lie on opposite sides of the tangent x = 0 at the origin. OO O 2a ()a ()b ()c 11 1 1 1 1 1 2 XX X YY Y Figure 9.7. Examples of a cusp of the first kind (a), a cusp of the second kind (b), and an osculation point (c). 9.1. THEORY OF CURVES 373 Example 10. For the curve (Fig. 9.7b) (y 2 – x 2 ) 2 – x 5 = 0, the origin is a cusp of the second kind, which easily follows from equation y = x 2 x 5/2 of the curve. It is also obvious that the curve consists of two branches tangent to the axis OX at the origin, and for 0 < x < 1 the value of y is positive for both branches. Example 11. The curve (Fig. 9.7c) y 2 – x 4 = 0 has an osculation point at the origin. Remark 1. If all the second partial derivatives are zero at the point M 0 , i.e., F xx = F xy = F yy = 0,then more than two branches of the curve can pass through this point. For example, for the trefoil (x 2 + y 2 ) 2 – ax(x 2 – y 2 )=0, three branches with tangents x = 0 and x y = 0 pass through the origin (Fig. 9.8). O a X Y Figure 9.8. The trefoil. Remark 2. If the equation F (x, y)=0 does not contain constant terms and terms of degree 1, then the origin is a double point. The equation of the tangent at a double point can readily be obtained by equating all terms of degree 2 with zero. For example, for the cissoid of Diocles (Example 9), the equation of the tangent x = 0 follows from the equation –xy 2 – x 3 = 0. If the equation F (x, y)=0 does not contain constant terms and terms of degrees 1 and 2, then the origin is a triple point, etc. Along with the singular points listed above, there are many other singular points with specific names: 1. Break points are points at which the curve changes its direction by a “jump” and, in contrast to cusps, the tangents to both parts of the curve are distinct (Fig. 9.9a). 2. Termination points are points at which the curve terminates (Fig. 9.9b). 3. Asymptotic points are points around which the curve winds infinitely many times while infinitely approaching them (Fig. 9.9c). ()a ()b ()c Figure 9.9. The break point (a), the termination point (b), and the asymptotic point (c). Remark. A break point corresponds to a jump discontinuity of the derivative dy/dx. Termination points correspond to either a jump discontinuity or termination of the function y = f(x). Asymptotic points can be found most easily in curves given in the polar coordinate system. . between the positive direction of the OX-axis and the positive direction of the tangent (Fig. 9.2a). The slope of the tangent (and the angle α) is determined by the formulas tan α = y x =– F x F y = y t x t . 9.1 evaluated at x = x 0 , y = y 0 ,andt = t 0 . The positive sense of the tangent coincides with the positive sense of the curve at the point of tangency; and the positive sense of the normal can in some. equations of the form F (x, y)=0 are used as a rule to find singular points of a curve and analyze their character. At any singular point M 0 (x 0 , y 0 ), both partial derivatives of the function